{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github"
      },
      "source": [
        "\u003ca href=\"https://colab.research.google.com/github/google/neural-tangents/blob/main/notebooks/learning_curves.ipynb\" target=\"_parent\"\u003e\u003cimg src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/\u003e\u003c/a\u003e"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "rMF0I5Sdl7GV"
      },
      "source": [
        "# Precise Learning Curves and Higher-Order Scaling Limits for Dot Product Kernel Regression\n",
        "\n",
        "Here we present a notebook that goes through the essential results in  [Xiao-Hu-Misiakiewicz-Lu-Pennington](https://arxiv.org/abs/2205.14846).\n",
        "\n",
        "**Acknowledgement**. We thank Roman Novak for providing valuable feedback for this notebook, which makes the notebook 20x faster to run.  "
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cIkq2O7yVWYJ"
      },
      "source": [
        "Copyright 2022 Google LLC\n",
        "\n",
        "Licensed under the Apache License, Version 2.0 (the \"License\");\n",
        "you may not use this file except in compliance with the License.\n",
        "You may obtain a copy of the License at\n",
        "\n",
        "     https://www.apache.org/licenses/LICENSE-2.0\n",
        "\n",
        "Unless required by applicable law or agreed to in writing, software\n",
        "distributed under the License is distributed on an \"AS IS\" BASIS,\n",
        "WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n",
        "See the License for the specific language governing permissions and\n",
        "limitations under the License."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "aWMU1YkS5lPB"
      },
      "source": [
        "### Imports \u0026 Utils"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 16,
      "metadata": {
        "executionInfo": {
          "elapsed": 105,
          "status": "ok",
          "timestamp": 1670966799585,
          "user": {
            "displayName": "",
            "userId": ""
          },
          "user_tz": 300
        },
        "id": "-U2XtTJ-kma4",
        "outputId": "564691af-494d-4a1a-c47b-6c49c421e089"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "/bin/sh: line 1: pip: command not found\n",
            "/bin/sh: line 1: pip: command not found\n",
            "/bin/sh: line 1: pip: command not found\n"
          ]
        }
      ],
      "source": [
        "!pip install -q --upgrade pip\n",
        "!pip install -q --upgrade jax[cuda11_cudnn82] -f https://storage.googleapis.com/jax-releases/jax_cuda_releases.html\n",
        "!pip install -q git+https://www.github.com/google/neural-tangents"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 28,
      "metadata": {
        "executionInfo": {
          "elapsed": 54,
          "status": "ok",
          "timestamp": 1670966876171,
          "user": {
            "displayName": "",
            "userId": ""
          },
          "user_tz": 300
        },
        "id": "-1nTu8wtiK5w"
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import jax.numpy as jnp\n",
        "\n",
        "from jax import grad, jit, vmap\n",
        "from jax import lax\n",
        "from jax import config\n",
        "from jax import random\n",
        "\n",
        "import jax \n",
        "import time\n",
        "import scipy as sp \n",
        "from jax import scipy as jsp\n",
        "\n",
        "\n",
        "config.update('jax_enable_x64', True)\n",
        "\n",
        "from functools import partial\n",
        "import neural_tangents as nt\n",
        "from neural_tangents import stax"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 18,
      "metadata": {
        "executionInfo": {
          "elapsed": 56,
          "status": "ok",
          "timestamp": 1670966799961,
          "user": {
            "displayName": "",
            "userId": ""
          },
          "user_tz": 300
        },
        "id": "xjtNH1OL7vl8"
      },
      "outputs": [],
      "source": [
        "from IPython.display import set_matplotlib_formats\n",
        "set_matplotlib_formats('pdf', 'svg')\n",
        "import matplotlib.pyplot as plt\n",
        "import seaborn as sns\n",
        "sns.set_style(style='white')\n",
        "\n",
        "\n",
        "def format_plot(x='', y='', grid=True):  \n",
        "  ax = plt.gca()\n",
        "  plt.grid(grid)\n",
        "  plt.xlabel(x, fontsize=20)\n",
        "  plt.ylabel(y, fontsize=20)\n",
        "  \n",
        "\n",
        "def finalize_plot(shape=(1, 1)):\n",
        "  plt.gcf().set_size_inches(\n",
        "    shape[0] * 1.5 * plt.gcf().get_size_inches()[1], \n",
        "    shape[1] * 1.5 * plt.gcf().get_size_inches()[1])\n",
        "  plt.tight_layout()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "QNg2jyWuinMl"
      },
      "source": [
        "### Helper Functions I: Genganbauer polynomials and Marchenko-Pastur distribution"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 19,
      "metadata": {
        "colab": {
          "height": 391
        },
        "executionInfo": {
          "elapsed": 951,
          "status": "ok",
          "timestamp": 1670966801028,
          "user": {
            "displayName": "",
            "userId": ""
          },
          "user_tz": 300
        },
        "id": "iNS7Qrb7LHY2",
        "outputId": "65f26f00-86ca-4aa5-f7c8-98f687af9e21"
      },
      "outputs": [
        {
          "data": {
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TTTjLu0lSjBmUMuoepnYifaw8yKM66GRNzqwjmdnTT9uZ+Bxwt1/aZE6Vx3Qe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\nSXCMiVCRgHaB2kgMOXssdlE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            "text/plain": [
              "\u003cFigure size 600x400 with 1 Axes\u003e"
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          "output_type": "display_data"
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      "source": [
        "# Computing dimensions of spherical harmonics. \n",
        "def sh_dim(p, degree):\n",
        "  if degree == 0: return 1 \n",
        "  return int(\n",
        "      sp.special.comb(degree + p - 3, degree - 1) * \n",
        "      (2 * degree + p - 2) / degree)\n",
        "\n",
        "\n",
        "def sh_dim_sum(d, i):\n",
        "  if i == 0: return 1 \n",
        "  return sh_dim_sum(d, i - 1) + sh_dim(d, i)\n",
        "\n",
        "\n",
        "# Gegenbauer polynomials; see https://en.wikipedia.org/wiki/Gegenbauer_polynomials. \n",
        "def gegenbauer_fn(x, a, n):\n",
        "  if n == 0: return 1. \n",
        "  if n == 1: return 2 * a * x \n",
        "  return (\n",
        "      (2 * x * (n + a - 1)) * gegenbauer_fn(x, a, n - 1) - \n",
        "      (n + 2 * a - 2) * gegenbauer_fn(x, a, n - 2)\n",
        "      ) / n \n",
        "\n",
        "\n",
        "@partial(jit, static_argnums=(2,))\n",
        "def normalized_gegenbauer(x, a, n):\n",
        "  return gegenbauer_fn(x, a, n) / gegenbauer_fn(1., a, n)\n",
        "\n",
        "  \n",
        "# Marchenko-Pastur \n",
        "@jit \n",
        "def get_mp_density(alpha, n_points=1000): \n",
        "  a = (1 + jnp.sqrt(alpha)) ** 2\n",
        "  b = (1 - jnp.sqrt(alpha)) ** 2\n",
        "  def f(x):\n",
        "    return 1 / (2 * jnp.pi) * ((a - x) * (x - b)) **0.5 / x / alpha \n",
        "  y = jnp.linspace(b, a, n_points)\n",
        "  return y, f(y)\n",
        "\n",
        "\n",
        "# Spherical data \n",
        "def get_input(n, d, key=jax.random.PRNGKey(10)):\n",
        "  x = random.normal(key, (n, d))\n",
        "  x /=  jnp.linalg.norm(x, axis=-1, keepdims=True)\n",
        "  return x  \n",
        "\n",
        "\n",
        "for alpha in [0.3, 0.5, 0.7, 1.,]:\n",
        "  x, y = get_mp_density(alpha)\n",
        "  plt.plot(x, y, label=f'$\\\\alpha$ = {alpha:.2f}')\n",
        "plt.xlabel('Value')\n",
        "plt.ylabel('Density')\n",
        "plt.legend()\n",
        "plt.title('Marchenko-Pastur Distribution');\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "wlZlJGUQEg00"
      },
      "source": [
        "The first key result in the paper is that the spectrum of the Gram matrix of higher-order spherical harmonics, namely, $\\frac 1m Y_r(X)^\\top Y_r(X)$, follows the MP law.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 20,
      "metadata": {
        "colab": {
          "height": 373
        },
        "executionInfo": {
          "elapsed": 2973,
          "status": "ok",
          "timestamp": 1670966804169,
          "user": {
            "displayName": "",
            "userId": ""
          },
          "user_tz": 300
        },
        "id": "PYnSJ0PqU7tc",
        "outputId": "fa4f1f02-98ce-44dc-bff2-24ffc3730fcd"
      },
      "outputs": [
        {
          "data": {
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InKUP5peLs+ljHGI7LFz8gb+DiVmRlKHNwYILO4a/j+o8lItj7g0NBo2E+cSP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\nPMSC76vxHh75rMzJswfGL9l3Dyv21IRlIePFGdp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            "text/plain": [
              "\u003cFigure size 1500x400 with 3 Axes\u003e"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "colors = (plt.rcParams['axes.prop_cycle'].by_key()['color'])\n",
        "s = jax.random.PRNGKey(12)\n",
        "\n",
        "\n",
        "@partial(jit, static_argnums=2)\n",
        "def get_evals(x, d, degree):\n",
        "    nngp = normalized_gegenbauer(x@x.T, d / 2. - 1, degree) \n",
        "    factor = jnp.trace(nngp) / x.shape[0]\n",
        "    nngp /= factor\n",
        "    evals = jnp.linalg.eigvalsh(nngp)\n",
        "    return evals\n",
        "\n",
        "\n",
        "alphas = [0.2, 0.5, 1.]\n",
        "degrees = [2, 3, 4]\n",
        "dims =  [150, 30, 14]\n",
        "\n",
        "\n",
        "fig, axes = plt.subplots(1, 3, figsize=(5 * len(alphas), 4), sharey=False)\n",
        "\n",
        "\n",
        "for i, (alpha, degree, dim) in enumerate(zip(alphas, degrees, dims)):  \n",
        "  n_samples = int(sh_dim(dim, degree) * alpha)\n",
        "  key, s = random.split(s)\n",
        "  x = get_input(n_samples, dim, key=key) \n",
        "  evals = get_evals(x, dim, degree) \n",
        "  ax = axes[i]\n",
        "  sns.histplot(evals, label='Empirical', color=colors[i], alpha=0.5, bins=100, \n",
        "               ax=ax, stat=\"density\")\n",
        "  x, y = get_mp_density(alpha)\n",
        "  ax.plot(x, y, label=f'$\\\\mu_\\\\alpha$, $\\\\alpha=${alpha:.2f}', color=colors[i])\n",
        "  ax.set_title(f'deg={degree}, dim={dim}, n_sample={n_samples}')\n",
        "  ax.legend()\n",
        "    \n",
        "plt.show()\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "aJfqQhPHooPt"
      },
      "source": [
        "### Helper functions II: Kernels and Labels. "
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 21,
      "metadata": {
        "executionInfo": {
          "elapsed": 105,
          "status": "ok",
          "timestamp": 1670966804433,
          "user": {
            "displayName": "",
            "userId": ""
          },
          "user_tz": 300
        },
        "id": "zyZhNMBnUJbC"
      },
      "outputs": [],
      "source": [
        "def get_labels(x, label_degrees, energies, key=random.PRNGKey(20)):\n",
        "  \"\"\"We use monomials of the form `c1*x1 + c2*x1*x2 + c3*x1*x2*x3 + ... ` as our labels. \"\"\"\n",
        "  key, s = random.split(key)\n",
        "  labels = [] \n",
        "  n = x.shape[0]\n",
        "  for k, energy in zip(label_degrees, energies):\n",
        "    out = jnp.ones_like(x)\n",
        "    for l in range(k):\n",
        "      out = jnp.multiply(out, jnp.roll(x, l, axis=-1)) \n",
        "    w = random.normal(key, (x.shape[1], 1))\n",
        "    key, s = random.split(key)\n",
        "    out = out @ w  \n",
        "    out /= jnp.linalg.norm(out) \n",
        "    out *= jnp.sqrt(n)\n",
        "    zeta = energy ** 0.5 \n",
        "    labels += [out * zeta]\n",
        "  labels = jnp.array(labels) \n",
        "  final_labels = jnp.sum(labels, axis=0) \n",
        "  return final_labels \n",
        "\n",
        "\n",
        "def get_kernel(x1, \n",
        "               x2, \n",
        "               degrees=[1, 2, 3, 4, 5, 6, 7],  \n",
        "               evals=[1. / 2**i for i in range(7)], \n",
        "               ):\n",
        "  \"\"\"The kernel is constructed by summing Gegenbauer polynomials of various degrees.\"\"\"\n",
        "  n, d = x1.shape\n",
        "  x11 = x1@x1.T\n",
        "  x12 = x1@x2.T  \n",
        "  kernel = 0.\n",
        "\n",
        "  for degree, eval in zip(degrees, evals): \n",
        "    alpha = d / 2. - 1 \n",
        "    nngp = normalized_gegenbauer(x12, alpha, degree)\n",
        "    factor = jnp.trace(normalized_gegenbauer(x11, alpha, degree)) / n  \n",
        "    nngp /= factor\n",
        "    kernel += nngp * eval \n",
        "\n",
        "  return kernel\n",
        "\n",
        "\n",
        "def get_conv_labels(x,  \n",
        "                    label_degrees, \n",
        "                    energies,\n",
        "                    key=random.PRNGKey(20),\n",
        "                    ):\n",
        "  \n",
        "  _get = partial(get_labels,  \n",
        "                 label_degrees=label_degrees,\n",
        "                 energies=energies,\n",
        "                 key=key\n",
        "                 )\n",
        "  _get_labels = jax.vmap(_get, in_axes=1)\n",
        "  labels = _get_labels(x)\n",
        "  return jnp.sum(labels, axis=0) / jnp.sqrt(p)\n",
        "\n",
        "\n",
        "def get_conv_kernel(x1, \n",
        "                    x2, \n",
        "                    degrees=[1, 2, 3, 4, 5, 6, 7], \n",
        "                    evals=[1. / 2**i for i in range(10)]\n",
        "                    ):\n",
        "  _get_kernel = partial(get_kernel, degrees=degrees, evals=evals)\n",
        "  _helper = jax.vmap(_get_kernel, in_axes=(1, 1)) \n",
        "  kernel = _helper(x1, x2)\n",
        "  return jnp.mean(kernel, axis=0) "
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 36,
      "metadata": {
        "executionInfo": {
          "elapsed": 136,
          "status": "ok",
          "timestamp": 1670967114072,
          "user": {
            "displayName": "",
            "userId": ""
          },
          "user_tz": 300
        },
        "id": "V9x74oakFjSk"
      },
      "outputs": [],
      "source": [
        "\n",
        "def get_meta_loss(n=10000, \n",
        "                  d=60, \n",
        "                  p=1, \n",
        "                  kernel_degrees=[1, 2, 3, 4, 5, 6, 7], \n",
        "                  label_degrees=[1, 2, 3, 4, 5, 6, 7], \n",
        "                  energies=[1. / i for i in range(1, 10)], \n",
        "                  n_average=5, \n",
        "                  evals=[1./2**i for i in range(7)], \n",
        "                  key=random.PRNGKey(10)):\n",
        "  \n",
        "  ker_fn = get_kernel if p == 1 else get_conv_kernel\n",
        "  ker_fn = partial(ker_fn, degrees=kernel_degrees, evals=evals)\n",
        "  ker_fn = jit(ker_fn)\n",
        "\n",
        "  @jit \n",
        "  def get_mse(ktd, kdd, y_train, y_test):\n",
        "    residual = y_test - ktd @ jsp.linalg.solve(kdd, y_train, assume_a='pos')    \n",
        "    return jnp.mean(jnp.square(residual)) / 2 \n",
        "  \n",
        "  n_trains = list(np.unique((1.3 ** np.arange(1, 100)).astype(int))) + [n]\n",
        "  n_trains += [int((sh_dim_sum(d, i) - 1) * scale * p) for i in range(1, 7) for scale in [0.8, 0.9, 1., 1.1, 1.2]] \n",
        "  n_trains += [int(sh_dim(d, i) * p) for i in range(1, 7) ] + [int((sh_dim_sum(d, i)-1) * p) for i in range(1, 7)]\n",
        "  n_trains = list(set(n_trains))\n",
        "  n_trains = jnp.sort(jnp.array([n_train for n_train in n_trains if n_train \u003c= n])) \n",
        "\n",
        "  n_test = 10000 \n",
        "\n",
        "  losses = []\n",
        "  key, s = random.split(key)\n",
        "\n",
        "  if p == 1: \n",
        "    @jit \n",
        "    def get_input_label(key):\n",
        "      x_all = get_input(n + n_test, d, key=key)\n",
        "      y_all = get_labels(x_all, label_degrees, energies)\n",
        "      return x_all, y_all \n",
        "  else: \n",
        "    @jit \n",
        "    def get_input_label(key):\n",
        "      x_all = get_input((n + n_test) * p, d, key=key).reshape((n + 10000, p, d))\n",
        "      y_all = get_conv_labels(x_all, label_degrees, energies)\n",
        "      return x_all, y_all \n",
        "\n",
        "  for i in range(n_average):\n",
        "    start_time = time.time() \n",
        "    key, s = random.split(s)\n",
        "\n",
        "    x_all, y_all = get_input_label(key)\n",
        "    x_train, x_test = x_all[:n], x_all[n:]\n",
        "    y_train, y_test = y_all[:n], y_all[n:]\n",
        "\n",
        "    kdd = ker_fn(x_train, x_train)\n",
        "    ktd = ker_fn(x_test, x_train)\n",
        "\n",
        "    losses.append(\n",
        "        [get_mse(\n",
        "            ktd[:, :n_train], \n",
        "            kdd[:n_train, :n_train], \n",
        "            y_train[:n_train], \n",
        "            y_test) \n",
        "        for n_train in n_trains]\n",
        "        )\n",
        "    run_time = time.time() - start_time\n",
        "    print(f'The {i + 1} / {n_average} simulation takes {run_time:.2f} seconds.')\n",
        "  losses_array = jnp.array(losses)\n",
        "\n",
        "  return n_trains, losses_array\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-YLDRMt5F6mu"
      },
      "source": [
        "Setup some experiment parameters."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 37,
      "metadata": {
        "executionInfo": {
          "elapsed": 54,
          "status": "ok",
          "timestamp": 1670967115954,
          "user": {
            "displayName": "",
            "userId": ""
          },
          "user_tz": 300
        },
        "id": "tWAUX_qB_1JF"
      },
      "outputs": [],
      "source": [
        "n = 12000 # maximal number of training samples\n",
        "d = 24 # input dimension \n",
        "p = 1 # number of patches. p = 1 for fully-connected network kernels. \n",
        "n_average = 20\n",
        "eigen_gap = 32 # Spectral gap between eigenspaces. \n",
        "evals = [eigen_gap ** (-i) for i in range(7)]\n",
        "label_degrees = [1, 2, 3, 4, 5, 6, 7]\n",
        "kernel_degrees = label_degrees \n",
        "energies = [1. / (i**2) for i in range(1, 10)]\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "mxqwSI4t__zV"
      },
      "source": [
        "Simulating the empirical sample-wise learning curves."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 39,
      "metadata": {
        "colab": {
          "height": 756
        },
        "executionInfo": {
          "elapsed": 41106,
          "status": "ok",
          "timestamp": 1670967272715,
          "user": {
            "displayName": "",
            "userId": ""
          },
          "user_tz": 300
        },
        "id": "Cr7MFuCs1Y-3",
        "outputId": "2e7dbfc9-75dd-4833-e462-2c13803caaee"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "The 1 / 20 simulation takes 16.77 seconds.\n",
            "The 2 / 20 simulation takes 1.18 seconds.\n",
            "The 3 / 20 simulation takes 1.07 seconds.\n",
            "The 4 / 20 simulation takes 1.08 seconds.\n",
            "The 5 / 20 simulation takes 1.23 seconds.\n",
            "The 6 / 20 simulation takes 1.27 seconds.\n",
            "The 7 / 20 simulation takes 1.21 seconds.\n",
            "The 8 / 20 simulation takes 1.27 seconds.\n",
            "The 9 / 20 simulation takes 1.28 seconds.\n",
            "The 10 / 20 simulation takes 1.24 seconds.\n",
            "The 11 / 20 simulation takes 1.26 seconds.\n",
            "The 12 / 20 simulation takes 1.23 seconds.\n",
            "The 13 / 20 simulation takes 1.27 seconds.\n",
            "The 14 / 20 simulation takes 1.22 seconds.\n",
            "The 15 / 20 simulation takes 1.22 seconds.\n",
            "The 16 / 20 simulation takes 1.19 seconds.\n",
            "The 17 / 20 simulation takes 1.16 seconds.\n",
            "The 18 / 20 simulation takes 1.09 seconds.\n",
            "The 19 / 20 simulation takes 1.10 seconds.\n",
            "The 20 / 20 simulation takes 1.08 seconds.\n"
          ]
        },
        {
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yIC9GaWx0ZXIgL0ZsYXRlRGVjb2RlID4+CnN0cmVhbQp4nDMyt1AwULA0ARKG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\ngMYKZW5kc3RyZWFtCmVuZG9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$\\mathdefault{10^{3}}$ --\u003e\n      \u003cg style=\"fill: #262626\" transform=\"translate(274.041143 267.793937) scale(0.1 -0.1)\"\u003e\n       \u003cdefs\u003e\n        \u003cpath id=\"DejaVuSans-33\" d=\"M 2597 2516 \nQ 3050 2419 3304 2112 \nQ 3559 1806 3559 1356 \nQ 3559 666 3084 287 \nQ 2609 -91 1734 -91 \nQ 1441 -91 1130 -33 \nQ 819 25 488 141 \nL 488 750 \nQ 750 597 1062 519 \nQ 1375 441 1716 441 \nQ 2309 441 2620 675 \nQ 2931 909 2931 1356 \nQ 2931 1769 2642 2001 \nQ 2353 2234 1838 2234 \nL 1294 2234 \nL 1294 2753 \nL 1863 2753 \nQ 2328 2753 2575 2939 \nQ 2822 3125 2822 3475 \nQ 2822 3834 2567 4026 \nQ 2313 4219 1838 4219 \nQ 1578 4219 1281 4162 \nQ 984 4106 628 3988 \nL 628 4550 \nQ 988 4650 1302 4700 \nQ 1616 4750 1894 4750 \nQ 2613 4750 3031 4423 \nQ 3450 4097 3450 3541 \nQ 3450 3153 3228 2886 \nQ 3006 2619 2597 2516 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n       \u003c/defs\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-31\" transform=\"translate(0 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\nL 2419 1100 \nL 313 1100 \nL 313 1709 \nL 2253 4666 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n       \u003c/defs\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-31\" transform=\"translate(0 0.684375)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-30\" transform=\"translate(63.623047 0.684375)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-34\" transform=\"translate(128.203125 38.965625) scale(0.7)\"/\u003e\n      \u003c/g\u003e\n     \u003c/g\u003e\n    \u003c/g\u003e\n    \u003cg id=\"xtick_6\"/\u003e\n    \u003cg id=\"xtick_7\"/\u003e\n    \u003cg id=\"xtick_8\"/\u003e\n    \u003cg id=\"xtick_9\"/\u003e\n    \u003cg id=\"xtick_10\"/\u003e\n    \u003cg id=\"xtick_11\"/\u003e\n    \u003cg id=\"xtick_12\"/\u003e\n    \u003cg id=\"xtick_13\"/\u003e\n    \u003cg id=\"xtick_14\"/\u003e\n    \u003cg id=\"xtick_15\"/\u003e\n    \u003cg id=\"xtick_16\"/\u003e\n    \u003cg id=\"xtick_17\"/\u003e\n    \u003cg id=\"xtick_18\"/\u003e\n    \u003cg id=\"xtick_19\"/\u003e\n    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id=\"DejaVuSans-6d\" d=\"M 3328 2828 \nQ 3544 3216 3844 3400 \nQ 4144 3584 4550 3584 \nQ 5097 3584 5394 3201 \nQ 5691 2819 5691 2113 \nL 5691 0 \nL 5113 0 \nL 5113 2094 \nQ 5113 2597 4934 2840 \nQ 4756 3084 4391 3084 \nQ 3944 3084 3684 2787 \nQ 3425 2491 3425 1978 \nL 3425 0 \nL 2847 0 \nL 2847 2094 \nQ 2847 2600 2669 2842 \nQ 2491 3084 2119 3084 \nQ 1678 3084 1418 2786 \nQ 1159 2488 1159 1978 \nL 1159 0 \nL 581 0 \nL 581 3500 \nL 1159 3500 \nL 1159 2956 \nQ 1356 3278 1631 3431 \nQ 1906 3584 2284 3584 \nQ 2666 3584 2933 3390 \nQ 3200 3197 3328 2828 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n       \u003cpath id=\"DejaVuSans-3a\" d=\"M 750 794 \nL 1409 794 \nL 1409 0 \nL 750 0 \nL 750 794 \nz\nM 750 3309 \nL 1409 3309 \nL 1409 2516 \nL 750 2516 \nL 750 3309 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n       \u003cpath id=\"DejaVuSans-20\" transform=\"scale(0.015625)\"/\u003e\n       \u003cpath id=\"DejaVuSans-23\" d=\"M 3272 2816 \nL 2363 2816 \nL 2100 1772 \nL 3016 1772 \nL 3272 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xlink:href=\"#DejaVuSans-3a\" x=\"97.412109\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-20\" x=\"131.103516\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-23\" x=\"162.890625\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-20\" x=\"246.679688\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-54\" x=\"278.466797\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-72\" x=\"324.800781\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-61\" x=\"365.914062\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-69\" x=\"427.193359\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-6e\" x=\"454.976562\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-69\" x=\"518.355469\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-6e\" x=\"546.138672\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-67\" x=\"609.517578\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-20\" x=\"672.994141\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-53\" x=\"704.78125\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-61\" x=\"768.257812\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-6d\" x=\"829.537109\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-70\" x=\"926.949219\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-6c\" x=\"990.425781\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-65\" x=\"1018.208984\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-73\" x=\"1079.732422\"/\u003e\n      \u003cuse xlink:href=\"#DejaVuSans-2e\" x=\"1131.832031\"/\u003e\n     \u003c/g\u003e\n    \u003c/g\u003e\n   \u003c/g\u003e\n   \u003cg id=\"matplotlib.axis_2\"\u003e\n    \u003cg id=\"ytick_1\"\u003e\n     \u003cg id=\"line2d_6\"\u003e\n      \u003cpath d=\"M 43.78125 226.305035 \nL 378.58125 226.305035 \n\" clip-path=\"url(#pe94e11d334)\" style=\"fill: none; stroke: #cccccc; stroke-width: 0.8; stroke-linecap: round\"/\u003e\n     \u003c/g\u003e\n     \u003cg id=\"text_7\"\u003e\n      \u003c!-- 0.2 --\u003e\n      \u003cg style=\"fill: #262626\" 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            "text/plain": [
              "\u003cFigure size 600x400 with 1 Axes\u003e"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "all_trains, all_loss = get_meta_loss(n=n, \n",
        "                                     d=d, \n",
        "                                     p=p, \n",
        "                                     kernel_degrees=kernel_degrees, \n",
        "                                     label_degrees=label_degrees, \n",
        "                                     evals=evals, \n",
        "                                     energies=energies, \n",
        "                                     n_average=n_average)\n",
        "\n",
        "plt.errorbar(all_trains, \n",
        "             jnp.mean(all_loss, axis=0), \n",
        "             jnp.std(all_loss, axis=0),  \n",
        "             label='Simulation', \n",
        "             markersize=3, \n",
        "             fmt='o')\n",
        "\n",
        "plt.xscale('log')\n",
        "plt.grid()\n",
        "plt.title(f'Samplewise Learning Curve. \\n d = {d}, p = {p}')\n",
        "plt.xlabel(f'm: # Training Samples.')\n",
        "plt.ylabel('Test MSE')\n",
        "plt.legend();"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2uxxwApdq0nQ"
      },
      "source": [
        "### Analytical formulas for the bias and the variance.\n",
        "\n",
        "To obtain the closed-form formulas for the bias and the variance, we need to evaluate several integrals (see Sec. C5 in the appendix). Fortunately, we can leave the evluation to https://www.wolframalpha.com/. "
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 25,
      "metadata": {
        "executionInfo": {
          "elapsed": 55,
          "status": "ok",
          "timestamp": 1670966866669,
          "user": {
            "displayName": "",
            "userId": ""
          },
          "user_tz": 300
        },
        "id": "23de_HClZJqO"
      },
      "outputs": [],
      "source": [
        "# Eq (218) in Sec C.5\n",
        "def int_bc_1(b, c):\n",
        "  x = -1 + (1 + b + c) / (jnp.sqrt(b + b**2) + jnp.sqrt(c + c**2))\n",
        "  return jnp.pi * x \n",
        "\n",
        "\n",
        "# Eq (219) in Sec C.5\n",
        "def int_bc_2(b, c): \n",
        "  x = b + c + 2 * b * c + 2 * jnp.sqrt(b * c * (1 + b + c + b * c))\n",
        "  x *= 2 * jnp.sqrt(b**2 + b)\n",
        "  return jnp.pi / x \n",
        "\n",
        "\n",
        "# Eq (206) in C.5. \n",
        "def get_bias(N, m, eval):\n",
        "  gamma = eval * m / N \n",
        "  alpha = 1. * N / m \n",
        "  b_plus = (1 + jnp.sqrt(alpha)) ** 2\n",
        "  b_minus = (1 - jnp.sqrt(alpha)) ** 2\n",
        "  delta = b_plus - b_minus\n",
        "  b = (1 + gamma * b_minus) / delta / gamma \n",
        "  c = b_minus / delta \n",
        "  x = (delta * gamma) ** (-1) / (2 * jnp.pi * alpha * gamma) * int_bc_2(b, c)\n",
        "  x = jnp.where(alpha \u003e 1, x + (1 - 1. / alpha), x)\n",
        "  return x / 2\n",
        "\n",
        "\n",
        "# Eq (207) in C.5. \n",
        "def get_var(N, m, eval):\n",
        "  gamma = eval * m / N \n",
        "  alpha = 1. * N / m \n",
        "  b_plus = (1 + jnp.sqrt(alpha)) ** 2\n",
        "  b_minus = (1 - jnp.sqrt(alpha)) ** 2\n",
        "  delta = b_plus - b_minus\n",
        "  b = (1 + gamma * b_minus) / delta / gamma \n",
        "  c = b_minus / delta \n",
        "  x = 1. / (2 * jnp.pi * alpha * gamma) * int_bc_1(b, c)\n",
        "  x = jnp.where(alpha \u003e 1, x + (1 - 1. / alpha), x)\n",
        "  return (x / 2 - get_bias(N, m, eval)) * alpha * gamma * 2"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "vV8gMoESLbWo"
      },
      "source": [
        "### Analytical formula for the sample-wise learning curves. \n",
        "Here, we implement Eq (20) of the paper. "
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 40,
      "metadata": {
        "executionInfo": {
          "elapsed": 53,
          "status": "ok",
          "timestamp": 1670967314363,
          "user": {
            "displayName": "",
            "userId": ""
          },
          "user_tz": 300
        },
        "id": "u4e4j-wvO315"
      },
      "outputs": [],
      "source": [
        "all_train = jnp.array(all_trains)\n",
        "\n",
        "samples = jnp.geomspace(all_trains[0], all_trains[-1], 200)\n",
        "samples = jnp.unique(jnp.sort(jnp.concatenate([samples, all_trains])))\n",
        "\n",
        "\n",
        "def get_exact_prediction(\n",
        "    evals=evals, \n",
        "    energies=energies, \n",
        "    label_degrees=label_degrees, \n",
        "    kernel_degrees=kernel_degrees):\n",
        "  length = min(len(evals), len(energies), len(label_degrees)) - 1\n",
        "  n_labels = len(label_degrees)\n",
        "  n_kernels = len(kernel_degrees)\n",
        "\n",
        "  @jit \n",
        "  def _prediction(m):\n",
        "    bs = 0 \n",
        "    vs = 0\n",
        "    for i in range(length):\n",
        "      reg = evals[i] / sum(evals[i+1: n_kernels])\n",
        "      # Note that we compute the number of spherical harmonics with degree \u003c= i. \n",
        "      N = sum([sh_dim(d, label_degrees[j]) for j in range(0, i + 1)]) * p \n",
        "      bs += energies[i] * get_bias(N, m , reg)\n",
        "      vs += (1 + (get_var(N, m , reg))) * sum(energies[i+1:n_labels]) / 2 \n",
        "\n",
        "    correction = 0\n",
        "    for j in range(1, length):\n",
        "      correction -= sum(energies[j: n_labels]) / 2 \n",
        "    return bs, bs + vs + correction \n",
        "\n",
        "  return jnp.array([_prediction(m) for m in samples]).T"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 45,
      "metadata": {
        "colab": {
          "height": 481
        },
        "executionInfo": {
          "elapsed": 2330,
          "status": "ok",
          "timestamp": 1670967364474,
          "user": {
            "displayName": "",
            "userId": ""
          },
          "user_tz": 300
        },
        "id": "nboE8z4Zt7Tf",
        "outputId": "5951a2b4-7d4f-4d2c-ec9e-bf912711a193"
      },
      "outputs": [
        {
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gMzkwIDM5MCA1MDAgODM4IDMxOCAzNjEgMzE4IDMzNyA2MzYgNjM2IDYzNiA2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\nBb13Lv4KZW5kc3RyZWFtCmVuZG9iago2OSAwIG9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\u003c/g\u003e\n     \u003cg id=\"text_3\"\u003e\n      \u003c!-- $pN(d,\\leq 2)$ --\u003e\n      \u003cg style=\"fill: #262626\" transform=\"translate(289.965102 322.1555) scale(0.1 -0.1)\"\u003e\n       \u003cdefs\u003e\n        \u003cpath id=\"DejaVuSans-32\" d=\"M 1228 531 \nL 3431 531 \nL 3431 0 \nL 469 0 \nL 469 531 \nQ 828 903 1448 1529 \nQ 2069 2156 2228 2338 \nQ 2531 2678 2651 2914 \nQ 2772 3150 2772 3378 \nQ 2772 3750 2511 3984 \nQ 2250 4219 1831 4219 \nQ 1534 4219 1204 4116 \nQ 875 4013 500 3803 \nL 500 4441 \nQ 881 4594 1212 4672 \nQ 1544 4750 1819 4750 \nQ 2544 4750 2975 4387 \nQ 3406 4025 3406 3419 \nQ 3406 3131 3298 2873 \nQ 3191 2616 2906 2266 \nQ 2828 2175 2409 1742 \nQ 1991 1309 1228 531 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n       \u003c/defs\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-Oblique-70\" transform=\"translate(0 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-Oblique-4e\" transform=\"translate(63.476562 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-28\" transform=\"translate(138.28125 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-Oblique-64\" transform=\"translate(177.294922 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-2c\" transform=\"translate(240.771484 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-2264\" transform=\"translate(311.523438 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-32\" transform=\"translate(414.794922 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-29\" transform=\"translate(478.417969 0.015625)\"/\u003e\n      \u003c/g\u003e\n     \u003c/g\u003e\n    \u003c/g\u003e\n    \u003cg id=\"xtick_4\"\u003e\n     \u003cg id=\"line2d_4\"\u003e\n      \u003cpath d=\"M 410.116457 307.5555 \nL 410.116457 35.7555 \n\" clip-path=\"url(#p8fb20ae2ca)\" style=\"fill: none; stroke: #cccccc; stroke-width: 0.8; stroke-linecap: round\"/\u003e\n     \u003c/g\u003e\n     \u003cg id=\"text_4\"\u003e\n      \u003c!-- $pN(d,\\leq 3)$ --\u003e\n      \u003cg style=\"fill: #262626\" transform=\"translate(384.216457 322.1555) scale(0.1 -0.1)\"\u003e\n       \u003cdefs\u003e\n        \u003cpath id=\"DejaVuSans-33\" d=\"M 2597 2516 \nQ 3050 2419 3304 2112 \nQ 3559 1806 3559 1356 \nQ 3559 666 3084 287 \nQ 2609 -91 1734 -91 \nQ 1441 -91 1130 -33 \nQ 819 25 488 141 \nL 488 750 \nQ 750 597 1062 519 \nQ 1375 441 1716 441 \nQ 2309 441 2620 675 \nQ 2931 909 2931 1356 \nQ 2931 1769 2642 2001 \nQ 2353 2234 1838 2234 \nL 1294 2234 \nL 1294 2753 \nL 1863 2753 \nQ 2328 2753 2575 2939 \nQ 2822 3125 2822 3475 \nQ 2822 3834 2567 4026 \nQ 2313 4219 1838 4219 \nQ 1578 4219 1281 4162 \nQ 984 4106 628 3988 \nL 628 4550 \nQ 988 4650 1302 4700 \nQ 1616 4750 1894 4750 \nQ 2613 4750 3031 4423 \nQ 3450 4097 3450 3541 \nQ 3450 3153 3228 2886 \nQ 3006 2619 2597 2516 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n       \u003c/defs\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-Oblique-70\" transform=\"translate(0 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-Oblique-4e\" transform=\"translate(63.476562 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-28\" transform=\"translate(138.28125 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-Oblique-64\" transform=\"translate(177.294922 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-2c\" transform=\"translate(240.771484 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-2264\" transform=\"translate(311.523438 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-33\" transform=\"translate(414.794922 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-29\" transform=\"translate(478.417969 0.015625)\"/\u003e\n      \u003c/g\u003e\n     \u003c/g\u003e\n    \u003c/g\u003e\n    \u003cg id=\"xtick_5\"\u003e\n     \u003cg id=\"line2d_5\"\u003e\n      \u003cpath d=\"M 493.478125 307.5555 \nL 493.478125 35.7555 \n\" clip-path=\"url(#p8fb20ae2ca)\" style=\"fill: none; stroke: #cccccc; stroke-width: 0.8; stroke-linecap: round\"/\u003e\n     \u003c/g\u003e\n     \u003cg id=\"text_5\"\u003e\n      \u003c!-- $pN(d,\\leq 4)$ --\u003e\n      \u003cg style=\"fill: #262626\" transform=\"translate(467.578125 322.1555) scale(0.1 -0.1)\"\u003e\n       \u003cdefs\u003e\n        \u003cpath id=\"DejaVuSans-34\" d=\"M 2419 4116 \nL 825 1625 \nL 2419 1625 \nL 2419 4116 \nz\nM 2253 4666 \nL 3047 4666 \nL 3047 1625 \nL 3713 1625 \nL 3713 1100 \nL 3047 1100 \nL 3047 0 \nL 2419 0 \nL 2419 1100 \nL 313 1100 \nL 313 1709 \nL 2253 4666 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n       \u003c/defs\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-Oblique-70\" transform=\"translate(0 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-Oblique-4e\" transform=\"translate(63.476562 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-28\" transform=\"translate(138.28125 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-Oblique-64\" transform=\"translate(177.294922 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-2c\" transform=\"translate(240.771484 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-2264\" transform=\"translate(311.523438 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-34\" transform=\"translate(414.794922 0.015625)\"/\u003e\n       \u003cuse xlink:href=\"#DejaVuSans-29\" transform=\"translate(478.417969 0.015625)\"/\u003e\n      \u003c/g\u003e\n     \u003c/g\u003e\n    \u003c/g\u003e\n    \u003cg id=\"xtick_6\"/\u003e\n    \u003cg id=\"xtick_7\"/\u003e\n    \u003cg id=\"xtick_8\"/\u003e\n    \u003cg id=\"xtick_9\"/\u003e\n    \u003cg id=\"xtick_10\"/\u003e\n    \u003cg id=\"xtick_11\"/\u003e\n    \u003cg id=\"xtick_12\"/\u003e\n    \u003cg id=\"xtick_13\"/\u003e\n    \u003cg id=\"xtick_14\"/\u003e\n    \u003cg id=\"xtick_15\"/\u003e\n    \u003cg id=\"xtick_16\"/\u003e\n    \u003cg id=\"xtick_17\"/\u003e\n    \u003cg id=\"xtick_18\"/\u003e\n    \u003cg id=\"xtick_19\"/\u003e\n    \u003cg id=\"xtick_20\"/\u003e\n    \u003cg id=\"xtick_21\"/\u003e\n    \u003cg id=\"xtick_22\"/\u003e\n    \u003cg id=\"xtick_23\"/\u003e\n    \u003cg id=\"xtick_24\"/\u003e\n    \u003cg id=\"xtick_25\"/\u003e\n    \u003cg id=\"xtick_26\"/\u003e\n    \u003cg id=\"xtick_27\"/\u003e\n    \u003cg id=\"xtick_28\"/\u003e\n    \u003cg id=\"xtick_29\"/\u003e\n    \u003cg id=\"xtick_30\"/\u003e\n    \u003cg id=\"xtick_31\"/\u003e\n    \u003cg id=\"xtick_32\"/\u003e\n    \u003cg id=\"xtick_33\"/\u003e\n    \u003cg id=\"xtick_34\"/\u003e\n    \u003cg id=\"xtick_35\"/\u003e\n    \u003cg id=\"xtick_36\"/\u003e\n    \u003cg id=\"xtick_37\"/\u003e\n    \u003cg id=\"xtick_38\"/\u003e\n    \u003cg id=\"xtick_39\"/\u003e\n    \u003cg id=\"xtick_40\"/\u003e\n    \u003cg id=\"xtick_41\"/\u003e\n    \u003cg id=\"text_6\"\u003e\n     \u003c!-- m: # Training Samples. --\u003e\n     \u003cg style=\"fill: #262626\" transform=\"translate(212.098437 335.853937) scale(0.1 -0.1)\"\u003e\n      \u003cdefs\u003e\n       \u003cpath id=\"DejaVuSans-6d\" d=\"M 3328 2828 \nQ 3544 3216 3844 3400 \nQ 4144 3584 4550 3584 \nQ 5097 3584 5394 3201 \nQ 5691 2819 5691 2113 \nL 5691 0 \nL 5113 0 \nL 5113 2094 \nQ 5113 2597 4934 2840 \nQ 4756 3084 4391 3084 \nQ 3944 3084 3684 2787 \nQ 3425 2491 3425 1978 \nL 3425 0 \nL 2847 0 \nL 2847 2094 \nQ 2847 2600 2669 2842 \nQ 2491 3084 2119 3084 \nQ 1678 3084 1418 2786 \nQ 1159 2488 1159 1978 \nL 1159 0 \nL 581 0 \nL 581 3500 \nL 1159 3500 \nL 1159 2956 \nQ 1356 3278 1631 3431 \nQ 1906 3584 2284 3584 \nQ 2666 3584 2933 3390 \nQ 3200 3197 3328 2828 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n       \u003cpath id=\"DejaVuSans-3a\" d=\"M 750 794 \nL 1409 794 \nL 1409 0 \nL 750 0 \nL 750 794 \nz\nM 750 3309 \nL 1409 3309 \nL 1409 2516 \nL 750 2516 \nL 750 3309 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n       \u003cpath id=\"DejaVuSans-20\" transform=\"scale(0.015625)\"/\u003e\n       \u003cpath id=\"DejaVuSans-23\" d=\"M 3272 2816 \nL 2363 2816 \nL 2100 1772 \nL 3016 1772 \nL 3272 2816 \nz\nM 2803 4594 \nL 2478 3297 \nL 3391 3297 \nL 3719 4594 \nL 4219 4594 \nL 3897 3297 \nL 4872 3297 \nL 4872 2816 \nL 3775 2816 \nL 3519 1772 \nL 4513 1772 \nL 4513 1294 \nL 3397 1294 \nL 3072 0 \nL 2572 0 \nL 2894 1294 \nL 1978 1294 \nL 1656 0 \nL 1153 0 \nL 1478 1294 \nL 494 1294 \nL 494 1772 \nL 1594 1772 \nL 1856 2816 \nL 850 2816 \nL 850 3297 \nL 1978 3297 \nL 2297 4594 \nL 2803 4594 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n       \u003cpath id=\"DejaVuSans-54\" d=\"M -19 4666 \nL 3928 4666 \nL 3928 4134 \nL 2272 4134 \nL 2272 0 \nL 1638 0 \nL 1638 4134 \nL -19 4134 \nL -19 4666 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n       \u003cpath id=\"DejaVuSans-72\" d=\"M 2631 2963 \nQ 2534 3019 2420 3045 \nQ 2306 3072 2169 3072 \nQ 1681 3072 1420 2755 \nQ 1159 2438 1159 1844 \nL 1159 0 \nL 581 0 \nL 581 3500 \nL 1159 3500 \nL 1159 2956 \nQ 1341 3275 1631 3429 \nQ 1922 3584 2338 3584 \nQ 2397 3584 2469 3576 \nQ 2541 3569 2628 3553 \nL 2631 2963 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n       \u003cpath id=\"DejaVuSans-61\" d=\"M 2194 1759 \nQ 1497 1759 1228 1600 \nQ 959 1441 959 1056 \nQ 959 750 1161 570 \nQ 1363 391 1709 391 \nQ 2188 391 2477 730 \nQ 2766 1069 2766 1631 \nL 2766 1759 \nL 2194 1759 \nz\nM 3341 1997 \nL 3341 0 \nL 2766 0 \nL 2766 531 \nQ 2569 213 2275 61 \nQ 1981 -91 1556 -91 \nQ 1019 -91 701 211 \nQ 384 513 384 1019 \nQ 384 1609 779 1909 \nQ 1175 2209 1959 2209 \nL 2766 2209 \nL 2766 2266 \nQ 2766 2663 2505 2880 \nQ 2244 3097 1772 3097 \nQ 1472 3097 1187 3025 \nQ 903 2953 641 2809 \nL 641 3341 \nQ 956 3463 1253 3523 \nQ 1550 3584 1831 3584 \nQ 2591 3584 2966 3190 \nQ 3341 2797 3341 1997 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n       \u003cpath id=\"DejaVuSans-69\" d=\"M 603 3500 \nL 1178 3500 \nL 1178 0 \nL 603 0 \nL 603 3500 \nz\nM 603 4863 \nL 1178 4863 \nL 1178 4134 \nL 603 4134 \nL 603 4863 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n       \u003cpath id=\"DejaVuSans-6e\" d=\"M 3513 2113 \nL 3513 0 \nL 2938 0 \nL 2938 2094 \nQ 2938 2591 2744 2837 \nQ 2550 3084 2163 3084 \nQ 1697 3084 1428 2787 \nQ 1159 2491 1159 1978 \nL 1159 0 \nL 581 0 \nL 581 3500 \nL 1159 3500 \nL 1159 2956 \nQ 1366 3272 1645 3428 \nQ 1925 3584 2291 3584 \nQ 2894 3584 3203 3211 \nQ 3513 2838 3513 2113 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n       \u003cpath id=\"DejaVuSans-67\" d=\"M 2906 1791 \nQ 2906 2416 2648 2759 \nQ 2391 3103 1925 3103 \nQ 1463 3103 1205 2759 \nQ 947 2416 947 1791 \nQ 947 1169 1205 825 \nQ 1463 481 1925 481 \nQ 2391 481 2648 825 \nQ 2906 1169 2906 1791 \nz\nM 3481 434 \nQ 3481 -459 3084 -895 \nQ 2688 -1331 1869 -1331 \nQ 1566 -1331 1297 -1286 \nQ 1028 -1241 775 -1147 \nL 775 -588 \nQ 1028 -725 1275 -790 \nQ 1522 -856 1778 -856 \nQ 2344 -856 2625 -561 \nQ 2906 -266 2906 331 \nL 2906 616 \nQ 2728 306 2450 153 \nQ 2172 0 1784 0 \nQ 1141 0 747 490 \nQ 353 981 353 1791 \nQ 353 2603 747 3093 \nQ 1141 3584 1784 3584 \nQ 2172 3584 2450 3431 \nQ 2728 3278 2906 2969 \nL 2906 3500 \nL 3481 3500 \nL 3481 434 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n       \u003cpath id=\"DejaVuSans-53\" d=\"M 3425 4513 \nL 3425 3897 \nQ 3066 4069 2747 4153 \nQ 2428 4238 2131 4238 \nQ 1616 4238 1336 4038 \nQ 1056 3838 1056 3469 \nQ 1056 3159 1242 3001 \nQ 1428 2844 1947 2747 \nL 2328 2669 \nQ 3034 2534 3370 2195 \nQ 3706 1856 3706 1288 \nQ 3706 609 3251 259 \nQ 2797 -91 1919 -91 \nQ 1588 -91 1214 -16 \nQ 841 59 441 206 \nL 441 856 \nQ 825 641 1194 531 \nQ 1563 422 1919 422 \nQ 2459 422 2753 634 \nQ 3047 847 3047 1241 \nQ 3047 1584 2836 1778 \nQ 2625 1972 2144 2069 \nL 1759 2144 \nQ 1053 2284 737 2584 \nQ 422 2884 422 3419 \nQ 422 4038 858 4394 \nQ 1294 4750 2059 4750 \nQ 2388 4750 2728 4690 \nQ 3069 4631 3425 4513 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n       \u003cpath id=\"DejaVuSans-70\" d=\"M 1159 525 \nL 1159 -1331 \nL 581 -1331 \nL 581 3500 \nL 1159 3500 \nL 1159 2969 \nQ 1341 3281 1617 3432 \nQ 1894 3584 2278 3584 \nQ 2916 3584 3314 3078 \nQ 3713 2572 3713 1747 \nQ 3713 922 3314 415 \nQ 2916 -91 2278 -91 \nQ 1894 -91 1617 61 \nQ 1341 213 1159 525 \nz\nM 3116 1747 \nQ 3116 2381 2855 2742 \nQ 2594 3103 2138 3103 \nQ 1681 3103 1420 2742 \nQ 1159 2381 1159 1747 \nQ 1159 1113 1420 752 \nQ 1681 391 2138 391 \nQ 2594 391 2855 752 \nQ 3116 1113 3116 1747 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n       \u003cpath id=\"DejaVuSans-6c\" d=\"M 603 4863 \nL 1178 4863 \nL 1178 0 \nL 603 0 \nL 603 4863 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n       \u003cpath id=\"DejaVuSans-65\" d=\"M 3597 1894 \nL 3597 1613 \nL 953 1613 \nQ 991 1019 1311 708 \nQ 1631 397 2203 397 \nQ 2534 397 2845 478 \nQ 3156 559 3463 722 \nL 3463 178 \nQ 3153 47 2828 -22 \nQ 2503 -91 2169 -91 \nQ 1331 -91 842 396 \nQ 353 884 353 1716 \nQ 353 2575 817 3079 \nQ 1281 3584 2069 3584 \nQ 2775 3584 3186 3129 \nQ 3597 2675 3597 1894 \nz\nM 3022 2063 \nQ 3016 2534 2758 2815 \nQ 2500 3097 2075 3097 \nQ 1594 3097 1305 2825 \nQ 1016 2553 972 2059 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square\"/\u003e\n   \u003c/g\u003e\n   \u003cg id=\"patch_5\"\u003e\n    \u003cpath d=\"M 47.078125 307.5555 \nL 493.478125 307.5555 \n\" style=\"fill: none; stroke: #262626; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/\u003e\n   \u003c/g\u003e\n   \u003cg id=\"patch_6\"\u003e\n    \u003cpath d=\"M 47.078125 35.7555 \nL 493.478125 35.7555 \n\" style=\"fill: none; stroke: #262626; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/\u003e\n   \u003c/g\u003e\n   \u003cg id=\"text_15\"\u003e\n    \u003c!-- Sample-wise Learning Curve.  --\u003e\n    \u003cg style=\"fill: #262626\" transform=\"translate(180.845312 16.318125) scale(0.12 -0.12)\"\u003e\n     \u003cdefs\u003e\n      \u003cpath id=\"DejaVuSans-2d\" d=\"M 313 2009 \nL 1997 2009 \nL 1997 1497 \nL 313 1497 \nL 313 2009 \nz\n\" transform=\"scale(0.015625)\"/\u003e\n      \u003cpath id=\"DejaVuSans-77\" d=\"M 269 3500 \nL 844 3500 \nL 1563 769 \nL 2278 3500 \nL 2956 3500 \nL 3675 769 \nL 4391 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            "text/plain": [
              "\u003cFigure size 800x500 with 1 Axes\u003e"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "plt.figure(figsize=(8, 5))\n",
        "b, bv = get_exact_prediction()\n",
        "plt.plot(samples, bv, label='Theory Prediction')\n",
        "plt.errorbar(all_trains , jnp.mean(all_loss, axis=0), jnp.std(all_loss, axis=0),  \n",
        "            label='Simulation', \n",
        "            markersize=3, \n",
        "            fmt='o')\n",
        "\n",
        "\n",
        "plt.xscale('log')\n",
        "energies = jnp.array(energies)\n",
        "critical_index = jnp.array([int(sh_dim(d, i)) for i in [1, 2, 3, 4]])\n",
        "\n",
        "xs = [1, p * sh_dim(d, 1), p * sh_dim_sum(d, 2), p*sh_dim_sum(d, 3), p * sh_dim_sum(d, 4)]\n",
        "\n",
        "\n",
        "labels = ['1', '$pN(d,\\\\leq 1)$', '$pN(d,\\\\leq 2)$', '$pN(d,\\\\leq 3)$','$pN(d,\\\\leq 4)$', ]\n",
        "plt.xticks(xs, labels)\n",
        "\n",
        "\n",
        "ys = [sum(energies[i:7]) / 2 for i in range(4)] + [0, 0.5, 1.]\n",
        "labels = ['$\\\\hat{F}^2_{\\geq 1}$',\n",
        "          '$\\\\hat{F}^2_{\\geq 2}$', \n",
        "          '$\\\\hat{F}^2_{\\geq 3}$', \n",
        "          '$\\\\hat{F}^2_{\\geq 4}$'] + [0, 0.5,  1.,]\n",
        "plt.yticks(ys, labels)\n",
        "plt.grid()\n",
        "plt.title(f'Sample-wise Learning Curve. \\n d = {d}, p = {p}')\n",
        "plt.xlabel(f'm: # Training Samples.')\n",
        "plt.ylabel('Test MSE')\n",
        "plt.legend()\n",
        "plt.show()\n"
      ]
    }
  ],
  "metadata": {
    "accelerator": "GPU",
    "colab": {
      "last_runtime": {
        "build_target": "//learning/deepmind/public/tools/ml_python:ml_notebook",
        "kind": "private"
      },
      "name": "Sample-wise Learning Curves.ipynb",
      "provenance": [
        {
          "file_id": "/piper/depot/google3/third_party/py/neural_tangents/notebooks/phase_diagram.ipynb",
          "timestamp": 1670428146336
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